Problem 41 - Algebra A Combined Function

Question

Without graphing, find the domain of each function. $$ f(x)=5 \sqrt{x-20}+1 $$

Step-by-Step Solution

Answer

The domain of the function is \( [20, \infty) \).

Step 1: Identify the Function Type

The function given is \( f(x) = 5 \sqrt{x - 20} + 1 \). This includes a square root. The function is defined where the expression inside the square root is non-negative, since the square root of a negative number is not defined in the real number system.

Step 2: Set the Expression Inside the Square Root to Non-negative

For the function to be defined, the expression inside the square root, \( x - 20 \), must be greater than or equal to zero.Mathematically, this is expressed as: \[ x - 20 \geq 0 \]

Step 3: Solve the Inequality

To solve the inequality \( x - 20 \geq 0 \), we add 20 to both sides to isolate \( x \):\[ x \geq 20 \]

Step 4: State the Domain of the Function

The domain of \( f(x) = 5 \sqrt{x-20}+1 \) is all \( x \) values that satisfy the inequality found in Step 3. Thus, the domain is \( x \geq 20 \). In interval notation, the domain is written as:\[ [20, \, \infty) \]

Key Concepts

Understanding the Square Root Function

A square root function features expressions where a variable is inside the square root symbol, like \( \sqrt{x-20} \) in our function \( f(x) = 5 \sqrt{x-20} + 1 \). The core property of square roots is that they only output non-negative numbers, assuming the input is non-negative. That is, a square root is undefined for negative inputs in the set of real numbers. This brings us to the rule: the expressions inside the square root must be equal to or greater than zero.

In our example, this means ensuring that \( x-20 \geq 0 \), which ensures all values are valid for calculation. So, square root functions often come with restrictions on their domains, ensuring the expression inside the root is non-negative.

Exploring Inequalities in Mathematics

Inequalities help us understand which values satisfy certain conditions. For our square root function, we have the inequality \( x-20 \geq 0 \). Here are some key points about inequalities and how they function:

• Inequalities express the relationship between two values or expressions, indicating that one is greater than, less than, or equal to the other.
• Common inequality symbols include \( \geq \) (greater than or equal to) and \( \leq \) (less than or equal to).
• When solving an inequality like \( x-20 \geq 0 \), you perform similar operations as you would in an equation, such as adding or subtracting the same value from both sides.

For instance, by adding 20 to both sides of \( x-20 \geq 0 \), you find \( x \geq 20 \).
This tells us that all \( x \) values greater than or equal to 20 satisfy the inequality, keeping the function defined.

Interval Notation Made Easy

Interval notation is a simplified way to represent sets of numbers that satisfy a particular condition or inequality. Let's break it down, using our example of finding the domain of \( f(x) = 5 \sqrt{x-20} + 1 \):

• An interval is defined by two numbers: a starting point and an endpoint. These numbers show where the interval begins and ends.
• In interval notation, brackets \([ \text{ ]} \) and parentheses \( \text{ )} \) indicate whether endpoints are included or not.
  • Use square brackets \([ \text{ ]} \) to include an endpoint.
  • Use parentheses \( \text{ )} \) to exclude an endpoint.

For our function, the condition \( x \geq 20 \) leads to the interval notation \([20, \infty)\).
This tells us that 20 is included, and numbers go up to infinity, which is always represented by a parenthesis because infinity is not a reachable endpoint.

Interval notation thus gives a quick way to understand the domain of a function.